On the Kumjian-Pask algebras with finite dimension

Given a higher-rank graph (or k-graph) Λ and a unital commutative ring R, the Kumjian Pask algebra KPR(Λ) is a universal R-algebra associated to Λ modulo specific relations. The Kumjian-Pask algebras are higher-rank generalization of Leavitt path algebras which have been greatly considered in recent years. In this note, we determine finite-dimensional Kumjian-Pask algebras by giving necessary and sufficient graph-theoretic conditions. Briefly, we show that a Kumjian-Pask algebra KPR(Λ) is finite-dimensional if and only if the underlying graph Λ is finite and contains no cycles. Our result generalizes the known characterization of Leavitt path algebras with finite dimension